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## G = C5×C42.2C22order 320 = 26·5

### Direct product of C5 and C42.2C22

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×C42.2C22
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C20 — C5×C8⋊C4 — C5×C42.2C22
 Lower central C1 — C22 — C2×C4 — C5×C42.2C22
 Upper central C1 — C2×C10 — C4×C20 — C5×C42.2C22

Generators and relations for C5×C42.2C22
G = < a,b,c,d,e | a5=b4=c4=1, d2=c, e2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc2, ebe-1=b-1, cd=dc, ece-1=b2c-1, ede-1=b-1c2d >

Subgroups: 98 in 60 conjugacy classes, 30 normal (14 characteristic)
C1, C2, C2 [×2], C4 [×5], C22, C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], C10, C10 [×2], C42, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C20 [×5], C2×C10, C8⋊C4 [×2], C42.C2, C40 [×4], C2×C20, C2×C20 [×2], C2×C20 [×2], C42.2C22, C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40 [×2], C5×C8⋊C4 [×2], C5×C42.C2, C5×C42.2C22
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C2×C4, D4 [×2], C10 [×3], C22⋊C4, C20 [×2], C2×C10, C4.10D4, C4≀C2 [×2], C2×C20, C5×D4 [×2], C42.2C22, C5×C22⋊C4, C5×C4.10D4, C5×C4≀C2 [×2], C5×C42.2C22

Smallest permutation representation of C5×C42.2C22
Regular action on 320 points
Generators in S320
(1 47 39 31 23)(2 48 40 32 24)(3 41 33 25 17)(4 42 34 26 18)(5 43 35 27 19)(6 44 36 28 20)(7 45 37 29 21)(8 46 38 30 22)(9 233 225 217 209)(10 234 226 218 210)(11 235 227 219 211)(12 236 228 220 212)(13 237 229 221 213)(14 238 230 222 214)(15 239 231 223 215)(16 240 232 224 216)(49 81 73 65 57)(50 82 74 66 58)(51 83 75 67 59)(52 84 76 68 60)(53 85 77 69 61)(54 86 78 70 62)(55 87 79 71 63)(56 88 80 72 64)(89 121 113 105 97)(90 122 114 106 98)(91 123 115 107 99)(92 124 116 108 100)(93 125 117 109 101)(94 126 118 110 102)(95 127 119 111 103)(96 128 120 112 104)(129 168 153 145 137)(130 161 154 146 138)(131 162 155 147 139)(132 163 156 148 140)(133 164 157 149 141)(134 165 158 150 142)(135 166 159 151 143)(136 167 160 152 144)(169 201 193 185 177)(170 202 194 186 178)(171 203 195 187 179)(172 204 196 188 180)(173 205 197 189 181)(174 206 198 190 182)(175 207 199 191 183)(176 208 200 192 184)(241 273 265 257 249)(242 274 266 258 250)(243 275 267 259 251)(244 276 268 260 252)(245 277 269 261 253)(246 278 270 262 254)(247 279 271 263 255)(248 280 272 264 256)(281 313 305 297 289)(282 314 306 298 290)(283 315 307 299 291)(284 316 308 300 292)(285 317 309 301 293)(286 318 310 302 294)(287 319 311 303 295)(288 320 312 304 296)
(1 95 247 175)(2 92 248 172)(3 89 241 169)(4 94 242 174)(5 91 243 171)(6 96 244 176)(7 93 245 173)(8 90 246 170)(9 313 168 81)(10 318 161 86)(11 315 162 83)(12 320 163 88)(13 317 164 85)(14 314 165 82)(15 319 166 87)(16 316 167 84)(17 97 249 177)(18 102 250 182)(19 99 251 179)(20 104 252 184)(21 101 253 181)(22 98 254 178)(23 103 255 183)(24 100 256 180)(25 105 257 185)(26 110 258 190)(27 107 259 187)(28 112 260 192)(29 109 261 189)(30 106 262 186)(31 111 263 191)(32 108 264 188)(33 113 265 193)(34 118 266 198)(35 115 267 195)(36 120 268 200)(37 117 269 197)(38 114 270 194)(39 119 271 199)(40 116 272 196)(41 121 273 201)(42 126 274 206)(43 123 275 203)(44 128 276 208)(45 125 277 205)(46 122 278 202)(47 127 279 207)(48 124 280 204)(49 209 281 129)(50 214 282 134)(51 211 283 131)(52 216 284 136)(53 213 285 133)(54 210 286 130)(55 215 287 135)(56 212 288 132)(57 217 289 137)(58 222 290 142)(59 219 291 139)(60 224 292 144)(61 221 293 141)(62 218 294 138)(63 223 295 143)(64 220 296 140)(65 225 297 145)(66 230 298 150)(67 227 299 147)(68 232 300 152)(69 229 301 149)(70 226 302 146)(71 231 303 151)(72 228 304 148)(73 233 305 153)(74 238 306 158)(75 235 307 155)(76 240 308 160)(77 237 309 157)(78 234 310 154)(79 239 311 159)(80 236 312 156)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)(145 147 149 151)(146 148 150 152)(153 155 157 159)(154 156 158 160)(161 163 165 167)(162 164 166 168)(169 171 173 175)(170 172 174 176)(177 179 181 183)(178 180 182 184)(185 187 189 191)(186 188 190 192)(193 195 197 199)(194 196 198 200)(201 203 205 207)(202 204 206 208)(209 211 213 215)(210 212 214 216)(217 219 221 223)(218 220 222 224)(225 227 229 231)(226 228 230 232)(233 235 237 239)(234 236 238 240)(241 243 245 247)(242 244 246 248)(249 251 253 255)(250 252 254 256)(257 259 261 263)(258 260 262 264)(265 267 269 271)(266 268 270 272)(273 275 277 279)(274 276 278 280)(281 283 285 287)(282 284 286 288)(289 291 293 295)(290 292 294 296)(297 299 301 303)(298 300 302 304)(305 307 309 311)(306 308 310 312)(313 315 317 319)(314 316 318 320)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136)(137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152)(153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176)(177 178 179 180 181 182 183 184)(185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208)(209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224)(225 226 227 228 229 230 231 232)(233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248)(249 250 251 252 253 254 255 256)(257 258 259 260 261 262 263 264)(265 266 267 268 269 270 271 272)(273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288)(289 290 291 292 293 294 295 296)(297 298 299 300 301 302 303 304)(305 306 307 308 309 310 311 312)(313 314 315 316 317 318 319 320)
(1 287 5 283)(2 212 6 216)(3 53 7 49)(4 130 8 134)(9 201 13 205)(10 278 14 274)(11 127 15 123)(12 44 16 48)(17 61 21 57)(18 138 22 142)(19 291 23 295)(20 224 24 220)(25 69 29 65)(26 146 30 150)(27 299 31 303)(28 232 32 228)(33 77 37 73)(34 154 38 158)(35 307 39 311)(36 240 40 236)(41 85 45 81)(42 161 46 165)(43 315 47 319)(50 174 54 170)(51 247 55 243)(52 92 56 96)(58 182 62 178)(59 255 63 251)(60 100 64 104)(66 190 70 186)(67 263 71 259)(68 108 72 112)(74 198 78 194)(75 271 79 267)(76 116 80 120)(82 206 86 202)(83 279 87 275)(84 124 88 128)(89 133 93 129)(90 282 94 286)(91 211 95 215)(97 141 101 137)(98 290 102 294)(99 219 103 223)(105 149 109 145)(106 298 110 302)(107 227 111 231)(113 157 117 153)(114 306 118 310)(115 235 119 239)(121 164 125 168)(122 314 126 318)(131 175 135 171)(132 244 136 248)(139 183 143 179)(140 252 144 256)(147 191 151 187)(148 260 152 264)(155 199 159 195)(156 268 160 272)(162 207 166 203)(163 276 167 280)(169 213 173 209)(172 288 176 284)(177 221 181 217)(180 296 184 292)(185 229 189 225)(188 304 192 300)(193 237 197 233)(196 312 200 308)(204 320 208 316)(210 246 214 242)(218 254 222 250)(226 262 230 258)(234 270 238 266)(241 285 245 281)(249 293 253 289)(257 301 261 297)(265 309 269 305)(273 317 277 313)

G:=sub<Sym(320)| (1,47,39,31,23)(2,48,40,32,24)(3,41,33,25,17)(4,42,34,26,18)(5,43,35,27,19)(6,44,36,28,20)(7,45,37,29,21)(8,46,38,30,22)(9,233,225,217,209)(10,234,226,218,210)(11,235,227,219,211)(12,236,228,220,212)(13,237,229,221,213)(14,238,230,222,214)(15,239,231,223,215)(16,240,232,224,216)(49,81,73,65,57)(50,82,74,66,58)(51,83,75,67,59)(52,84,76,68,60)(53,85,77,69,61)(54,86,78,70,62)(55,87,79,71,63)(56,88,80,72,64)(89,121,113,105,97)(90,122,114,106,98)(91,123,115,107,99)(92,124,116,108,100)(93,125,117,109,101)(94,126,118,110,102)(95,127,119,111,103)(96,128,120,112,104)(129,168,153,145,137)(130,161,154,146,138)(131,162,155,147,139)(132,163,156,148,140)(133,164,157,149,141)(134,165,158,150,142)(135,166,159,151,143)(136,167,160,152,144)(169,201,193,185,177)(170,202,194,186,178)(171,203,195,187,179)(172,204,196,188,180)(173,205,197,189,181)(174,206,198,190,182)(175,207,199,191,183)(176,208,200,192,184)(241,273,265,257,249)(242,274,266,258,250)(243,275,267,259,251)(244,276,268,260,252)(245,277,269,261,253)(246,278,270,262,254)(247,279,271,263,255)(248,280,272,264,256)(281,313,305,297,289)(282,314,306,298,290)(283,315,307,299,291)(284,316,308,300,292)(285,317,309,301,293)(286,318,310,302,294)(287,319,311,303,295)(288,320,312,304,296), (1,95,247,175)(2,92,248,172)(3,89,241,169)(4,94,242,174)(5,91,243,171)(6,96,244,176)(7,93,245,173)(8,90,246,170)(9,313,168,81)(10,318,161,86)(11,315,162,83)(12,320,163,88)(13,317,164,85)(14,314,165,82)(15,319,166,87)(16,316,167,84)(17,97,249,177)(18,102,250,182)(19,99,251,179)(20,104,252,184)(21,101,253,181)(22,98,254,178)(23,103,255,183)(24,100,256,180)(25,105,257,185)(26,110,258,190)(27,107,259,187)(28,112,260,192)(29,109,261,189)(30,106,262,186)(31,111,263,191)(32,108,264,188)(33,113,265,193)(34,118,266,198)(35,115,267,195)(36,120,268,200)(37,117,269,197)(38,114,270,194)(39,119,271,199)(40,116,272,196)(41,121,273,201)(42,126,274,206)(43,123,275,203)(44,128,276,208)(45,125,277,205)(46,122,278,202)(47,127,279,207)(48,124,280,204)(49,209,281,129)(50,214,282,134)(51,211,283,131)(52,216,284,136)(53,213,285,133)(54,210,286,130)(55,215,287,135)(56,212,288,132)(57,217,289,137)(58,222,290,142)(59,219,291,139)(60,224,292,144)(61,221,293,141)(62,218,294,138)(63,223,295,143)(64,220,296,140)(65,225,297,145)(66,230,298,150)(67,227,299,147)(68,232,300,152)(69,229,301,149)(70,226,302,146)(71,231,303,151)(72,228,304,148)(73,233,305,153)(74,238,306,158)(75,235,307,155)(76,240,308,160)(77,237,309,157)(78,234,310,154)(79,239,311,159)(80,236,312,156), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160)(161,163,165,167)(162,164,166,168)(169,171,173,175)(170,172,174,176)(177,179,181,183)(178,180,182,184)(185,187,189,191)(186,188,190,192)(193,195,197,199)(194,196,198,200)(201,203,205,207)(202,204,206,208)(209,211,213,215)(210,212,214,216)(217,219,221,223)(218,220,222,224)(225,227,229,231)(226,228,230,232)(233,235,237,239)(234,236,238,240)(241,243,245,247)(242,244,246,248)(249,251,253,255)(250,252,254,256)(257,259,261,263)(258,260,262,264)(265,267,269,271)(266,268,270,272)(273,275,277,279)(274,276,278,280)(281,283,285,287)(282,284,286,288)(289,291,293,295)(290,292,294,296)(297,299,301,303)(298,300,302,304)(305,307,309,311)(306,308,310,312)(313,315,317,319)(314,316,318,320), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232)(233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248)(249,250,251,252,253,254,255,256)(257,258,259,260,261,262,263,264)(265,266,267,268,269,270,271,272)(273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288)(289,290,291,292,293,294,295,296)(297,298,299,300,301,302,303,304)(305,306,307,308,309,310,311,312)(313,314,315,316,317,318,319,320), (1,287,5,283)(2,212,6,216)(3,53,7,49)(4,130,8,134)(9,201,13,205)(10,278,14,274)(11,127,15,123)(12,44,16,48)(17,61,21,57)(18,138,22,142)(19,291,23,295)(20,224,24,220)(25,69,29,65)(26,146,30,150)(27,299,31,303)(28,232,32,228)(33,77,37,73)(34,154,38,158)(35,307,39,311)(36,240,40,236)(41,85,45,81)(42,161,46,165)(43,315,47,319)(50,174,54,170)(51,247,55,243)(52,92,56,96)(58,182,62,178)(59,255,63,251)(60,100,64,104)(66,190,70,186)(67,263,71,259)(68,108,72,112)(74,198,78,194)(75,271,79,267)(76,116,80,120)(82,206,86,202)(83,279,87,275)(84,124,88,128)(89,133,93,129)(90,282,94,286)(91,211,95,215)(97,141,101,137)(98,290,102,294)(99,219,103,223)(105,149,109,145)(106,298,110,302)(107,227,111,231)(113,157,117,153)(114,306,118,310)(115,235,119,239)(121,164,125,168)(122,314,126,318)(131,175,135,171)(132,244,136,248)(139,183,143,179)(140,252,144,256)(147,191,151,187)(148,260,152,264)(155,199,159,195)(156,268,160,272)(162,207,166,203)(163,276,167,280)(169,213,173,209)(172,288,176,284)(177,221,181,217)(180,296,184,292)(185,229,189,225)(188,304,192,300)(193,237,197,233)(196,312,200,308)(204,320,208,316)(210,246,214,242)(218,254,222,250)(226,262,230,258)(234,270,238,266)(241,285,245,281)(249,293,253,289)(257,301,261,297)(265,309,269,305)(273,317,277,313)>;

G:=Group( (1,47,39,31,23)(2,48,40,32,24)(3,41,33,25,17)(4,42,34,26,18)(5,43,35,27,19)(6,44,36,28,20)(7,45,37,29,21)(8,46,38,30,22)(9,233,225,217,209)(10,234,226,218,210)(11,235,227,219,211)(12,236,228,220,212)(13,237,229,221,213)(14,238,230,222,214)(15,239,231,223,215)(16,240,232,224,216)(49,81,73,65,57)(50,82,74,66,58)(51,83,75,67,59)(52,84,76,68,60)(53,85,77,69,61)(54,86,78,70,62)(55,87,79,71,63)(56,88,80,72,64)(89,121,113,105,97)(90,122,114,106,98)(91,123,115,107,99)(92,124,116,108,100)(93,125,117,109,101)(94,126,118,110,102)(95,127,119,111,103)(96,128,120,112,104)(129,168,153,145,137)(130,161,154,146,138)(131,162,155,147,139)(132,163,156,148,140)(133,164,157,149,141)(134,165,158,150,142)(135,166,159,151,143)(136,167,160,152,144)(169,201,193,185,177)(170,202,194,186,178)(171,203,195,187,179)(172,204,196,188,180)(173,205,197,189,181)(174,206,198,190,182)(175,207,199,191,183)(176,208,200,192,184)(241,273,265,257,249)(242,274,266,258,250)(243,275,267,259,251)(244,276,268,260,252)(245,277,269,261,253)(246,278,270,262,254)(247,279,271,263,255)(248,280,272,264,256)(281,313,305,297,289)(282,314,306,298,290)(283,315,307,299,291)(284,316,308,300,292)(285,317,309,301,293)(286,318,310,302,294)(287,319,311,303,295)(288,320,312,304,296), (1,95,247,175)(2,92,248,172)(3,89,241,169)(4,94,242,174)(5,91,243,171)(6,96,244,176)(7,93,245,173)(8,90,246,170)(9,313,168,81)(10,318,161,86)(11,315,162,83)(12,320,163,88)(13,317,164,85)(14,314,165,82)(15,319,166,87)(16,316,167,84)(17,97,249,177)(18,102,250,182)(19,99,251,179)(20,104,252,184)(21,101,253,181)(22,98,254,178)(23,103,255,183)(24,100,256,180)(25,105,257,185)(26,110,258,190)(27,107,259,187)(28,112,260,192)(29,109,261,189)(30,106,262,186)(31,111,263,191)(32,108,264,188)(33,113,265,193)(34,118,266,198)(35,115,267,195)(36,120,268,200)(37,117,269,197)(38,114,270,194)(39,119,271,199)(40,116,272,196)(41,121,273,201)(42,126,274,206)(43,123,275,203)(44,128,276,208)(45,125,277,205)(46,122,278,202)(47,127,279,207)(48,124,280,204)(49,209,281,129)(50,214,282,134)(51,211,283,131)(52,216,284,136)(53,213,285,133)(54,210,286,130)(55,215,287,135)(56,212,288,132)(57,217,289,137)(58,222,290,142)(59,219,291,139)(60,224,292,144)(61,221,293,141)(62,218,294,138)(63,223,295,143)(64,220,296,140)(65,225,297,145)(66,230,298,150)(67,227,299,147)(68,232,300,152)(69,229,301,149)(70,226,302,146)(71,231,303,151)(72,228,304,148)(73,233,305,153)(74,238,306,158)(75,235,307,155)(76,240,308,160)(77,237,309,157)(78,234,310,154)(79,239,311,159)(80,236,312,156), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160)(161,163,165,167)(162,164,166,168)(169,171,173,175)(170,172,174,176)(177,179,181,183)(178,180,182,184)(185,187,189,191)(186,188,190,192)(193,195,197,199)(194,196,198,200)(201,203,205,207)(202,204,206,208)(209,211,213,215)(210,212,214,216)(217,219,221,223)(218,220,222,224)(225,227,229,231)(226,228,230,232)(233,235,237,239)(234,236,238,240)(241,243,245,247)(242,244,246,248)(249,251,253,255)(250,252,254,256)(257,259,261,263)(258,260,262,264)(265,267,269,271)(266,268,270,272)(273,275,277,279)(274,276,278,280)(281,283,285,287)(282,284,286,288)(289,291,293,295)(290,292,294,296)(297,299,301,303)(298,300,302,304)(305,307,309,311)(306,308,310,312)(313,315,317,319)(314,316,318,320), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232)(233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248)(249,250,251,252,253,254,255,256)(257,258,259,260,261,262,263,264)(265,266,267,268,269,270,271,272)(273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288)(289,290,291,292,293,294,295,296)(297,298,299,300,301,302,303,304)(305,306,307,308,309,310,311,312)(313,314,315,316,317,318,319,320), (1,287,5,283)(2,212,6,216)(3,53,7,49)(4,130,8,134)(9,201,13,205)(10,278,14,274)(11,127,15,123)(12,44,16,48)(17,61,21,57)(18,138,22,142)(19,291,23,295)(20,224,24,220)(25,69,29,65)(26,146,30,150)(27,299,31,303)(28,232,32,228)(33,77,37,73)(34,154,38,158)(35,307,39,311)(36,240,40,236)(41,85,45,81)(42,161,46,165)(43,315,47,319)(50,174,54,170)(51,247,55,243)(52,92,56,96)(58,182,62,178)(59,255,63,251)(60,100,64,104)(66,190,70,186)(67,263,71,259)(68,108,72,112)(74,198,78,194)(75,271,79,267)(76,116,80,120)(82,206,86,202)(83,279,87,275)(84,124,88,128)(89,133,93,129)(90,282,94,286)(91,211,95,215)(97,141,101,137)(98,290,102,294)(99,219,103,223)(105,149,109,145)(106,298,110,302)(107,227,111,231)(113,157,117,153)(114,306,118,310)(115,235,119,239)(121,164,125,168)(122,314,126,318)(131,175,135,171)(132,244,136,248)(139,183,143,179)(140,252,144,256)(147,191,151,187)(148,260,152,264)(155,199,159,195)(156,268,160,272)(162,207,166,203)(163,276,167,280)(169,213,173,209)(172,288,176,284)(177,221,181,217)(180,296,184,292)(185,229,189,225)(188,304,192,300)(193,237,197,233)(196,312,200,308)(204,320,208,316)(210,246,214,242)(218,254,222,250)(226,262,230,258)(234,270,238,266)(241,285,245,281)(249,293,253,289)(257,301,261,297)(265,309,269,305)(273,317,277,313) );

G=PermutationGroup([(1,47,39,31,23),(2,48,40,32,24),(3,41,33,25,17),(4,42,34,26,18),(5,43,35,27,19),(6,44,36,28,20),(7,45,37,29,21),(8,46,38,30,22),(9,233,225,217,209),(10,234,226,218,210),(11,235,227,219,211),(12,236,228,220,212),(13,237,229,221,213),(14,238,230,222,214),(15,239,231,223,215),(16,240,232,224,216),(49,81,73,65,57),(50,82,74,66,58),(51,83,75,67,59),(52,84,76,68,60),(53,85,77,69,61),(54,86,78,70,62),(55,87,79,71,63),(56,88,80,72,64),(89,121,113,105,97),(90,122,114,106,98),(91,123,115,107,99),(92,124,116,108,100),(93,125,117,109,101),(94,126,118,110,102),(95,127,119,111,103),(96,128,120,112,104),(129,168,153,145,137),(130,161,154,146,138),(131,162,155,147,139),(132,163,156,148,140),(133,164,157,149,141),(134,165,158,150,142),(135,166,159,151,143),(136,167,160,152,144),(169,201,193,185,177),(170,202,194,186,178),(171,203,195,187,179),(172,204,196,188,180),(173,205,197,189,181),(174,206,198,190,182),(175,207,199,191,183),(176,208,200,192,184),(241,273,265,257,249),(242,274,266,258,250),(243,275,267,259,251),(244,276,268,260,252),(245,277,269,261,253),(246,278,270,262,254),(247,279,271,263,255),(248,280,272,264,256),(281,313,305,297,289),(282,314,306,298,290),(283,315,307,299,291),(284,316,308,300,292),(285,317,309,301,293),(286,318,310,302,294),(287,319,311,303,295),(288,320,312,304,296)], [(1,95,247,175),(2,92,248,172),(3,89,241,169),(4,94,242,174),(5,91,243,171),(6,96,244,176),(7,93,245,173),(8,90,246,170),(9,313,168,81),(10,318,161,86),(11,315,162,83),(12,320,163,88),(13,317,164,85),(14,314,165,82),(15,319,166,87),(16,316,167,84),(17,97,249,177),(18,102,250,182),(19,99,251,179),(20,104,252,184),(21,101,253,181),(22,98,254,178),(23,103,255,183),(24,100,256,180),(25,105,257,185),(26,110,258,190),(27,107,259,187),(28,112,260,192),(29,109,261,189),(30,106,262,186),(31,111,263,191),(32,108,264,188),(33,113,265,193),(34,118,266,198),(35,115,267,195),(36,120,268,200),(37,117,269,197),(38,114,270,194),(39,119,271,199),(40,116,272,196),(41,121,273,201),(42,126,274,206),(43,123,275,203),(44,128,276,208),(45,125,277,205),(46,122,278,202),(47,127,279,207),(48,124,280,204),(49,209,281,129),(50,214,282,134),(51,211,283,131),(52,216,284,136),(53,213,285,133),(54,210,286,130),(55,215,287,135),(56,212,288,132),(57,217,289,137),(58,222,290,142),(59,219,291,139),(60,224,292,144),(61,221,293,141),(62,218,294,138),(63,223,295,143),(64,220,296,140),(65,225,297,145),(66,230,298,150),(67,227,299,147),(68,232,300,152),(69,229,301,149),(70,226,302,146),(71,231,303,151),(72,228,304,148),(73,233,305,153),(74,238,306,158),(75,235,307,155),(76,240,308,160),(77,237,309,157),(78,234,310,154),(79,239,311,159),(80,236,312,156)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120),(121,123,125,127),(122,124,126,128),(129,131,133,135),(130,132,134,136),(137,139,141,143),(138,140,142,144),(145,147,149,151),(146,148,150,152),(153,155,157,159),(154,156,158,160),(161,163,165,167),(162,164,166,168),(169,171,173,175),(170,172,174,176),(177,179,181,183),(178,180,182,184),(185,187,189,191),(186,188,190,192),(193,195,197,199),(194,196,198,200),(201,203,205,207),(202,204,206,208),(209,211,213,215),(210,212,214,216),(217,219,221,223),(218,220,222,224),(225,227,229,231),(226,228,230,232),(233,235,237,239),(234,236,238,240),(241,243,245,247),(242,244,246,248),(249,251,253,255),(250,252,254,256),(257,259,261,263),(258,260,262,264),(265,267,269,271),(266,268,270,272),(273,275,277,279),(274,276,278,280),(281,283,285,287),(282,284,286,288),(289,291,293,295),(290,292,294,296),(297,299,301,303),(298,300,302,304),(305,307,309,311),(306,308,310,312),(313,315,317,319),(314,316,318,320)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136),(137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152),(153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176),(177,178,179,180,181,182,183,184),(185,186,187,188,189,190,191,192),(193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208),(209,210,211,212,213,214,215,216),(217,218,219,220,221,222,223,224),(225,226,227,228,229,230,231,232),(233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248),(249,250,251,252,253,254,255,256),(257,258,259,260,261,262,263,264),(265,266,267,268,269,270,271,272),(273,274,275,276,277,278,279,280),(281,282,283,284,285,286,287,288),(289,290,291,292,293,294,295,296),(297,298,299,300,301,302,303,304),(305,306,307,308,309,310,311,312),(313,314,315,316,317,318,319,320)], [(1,287,5,283),(2,212,6,216),(3,53,7,49),(4,130,8,134),(9,201,13,205),(10,278,14,274),(11,127,15,123),(12,44,16,48),(17,61,21,57),(18,138,22,142),(19,291,23,295),(20,224,24,220),(25,69,29,65),(26,146,30,150),(27,299,31,303),(28,232,32,228),(33,77,37,73),(34,154,38,158),(35,307,39,311),(36,240,40,236),(41,85,45,81),(42,161,46,165),(43,315,47,319),(50,174,54,170),(51,247,55,243),(52,92,56,96),(58,182,62,178),(59,255,63,251),(60,100,64,104),(66,190,70,186),(67,263,71,259),(68,108,72,112),(74,198,78,194),(75,271,79,267),(76,116,80,120),(82,206,86,202),(83,279,87,275),(84,124,88,128),(89,133,93,129),(90,282,94,286),(91,211,95,215),(97,141,101,137),(98,290,102,294),(99,219,103,223),(105,149,109,145),(106,298,110,302),(107,227,111,231),(113,157,117,153),(114,306,118,310),(115,235,119,239),(121,164,125,168),(122,314,126,318),(131,175,135,171),(132,244,136,248),(139,183,143,179),(140,252,144,256),(147,191,151,187),(148,260,152,264),(155,199,159,195),(156,268,160,272),(162,207,166,203),(163,276,167,280),(169,213,173,209),(172,288,176,284),(177,221,181,217),(180,296,184,292),(185,229,189,225),(188,304,192,300),(193,237,197,233),(196,312,200,308),(204,320,208,316),(210,246,214,242),(218,254,222,250),(226,262,230,258),(234,270,238,266),(241,285,245,281),(249,293,253,289),(257,301,261,297),(265,309,269,305),(273,317,277,313)])

95 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 5A 5B 5C 5D 8A ··· 8H 10A ··· 10L 20A ··· 20P 20Q 20R 20S 20T 20U ··· 20AB 40A ··· 40AF order 1 2 2 2 4 4 4 4 4 4 4 5 5 5 5 8 ··· 8 10 ··· 10 20 ··· 20 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 2 2 4 8 8 1 1 1 1 4 ··· 4 1 ··· 1 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

95 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + - image C1 C2 C2 C4 C5 C10 C10 C20 D4 C4≀C2 C5×D4 C5×C4≀C2 C4.10D4 C5×C4.10D4 kernel C5×C42.2C22 C5×C8⋊C4 C5×C42.C2 C5×C4⋊C4 C42.2C22 C8⋊C4 C42.C2 C4⋊C4 C2×C20 C10 C2×C4 C2 C10 C2 # reps 1 2 1 4 4 8 4 16 2 8 8 32 1 4

Matrix representation of C5×C42.2C22 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 37 0 0 0 0 37
,
 32 39 0 0 0 9 0 0 0 0 40 9 0 0 18 1
,
 32 0 0 0 0 32 0 0 0 0 32 40 0 0 39 9
,
 22 35 0 0 7 19 0 0 0 0 0 37 0 0 33 31
,
 34 35 0 0 22 7 0 0 0 0 25 7 0 0 28 16
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,37,0,0,0,0,37],[32,0,0,0,39,9,0,0,0,0,40,18,0,0,9,1],[32,0,0,0,0,32,0,0,0,0,32,39,0,0,40,9],[22,7,0,0,35,19,0,0,0,0,0,33,0,0,37,31],[34,22,0,0,35,7,0,0,0,0,25,28,0,0,7,16] >;

C5×C42.2C22 in GAP, Magma, Sage, TeX

C_5\times C_4^2._2C_2^2
% in TeX

G:=Group("C5xC4^2.2C2^2");
// GroupNames label

G:=SmallGroup(320,135);
// by ID

G=gap.SmallGroup(320,135);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,2803,2530,248,4911,102]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=c^4=1,d^2=c,e^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^2,e*b*e^-1=b^-1,c*d=d*c,e*c*e^-1=b^2*c^-1,e*d*e^-1=b^-1*c^2*d>;
// generators/relations

׿
×
𝔽